Econ 407: Production Functions: Below are examples of three production functions. The firt one is linear and exhibits constant returns to sacle. This means if you double the input--x, the output y1 will double. The second function y2 shows increasing returns to scale; that is, if you double x, then the output more than doubles. The third function shows decreasing returns to sacle--doubling input does not result in doubling output. The graph of all the functions are also shown. Note the difference between the green (increasing returns), the red (constant returns) and the yellow (decreasing returns) to scale.

> y1:=5*x;

> y2:=5*x^2;

> y3:=5*x^0.5;

> plot({y1,y2,y3},x=0..2.5);

One of the most famous production functions used in economics is the Cobb-Douglas production function-- named after the two professors who developed it. This is a constant returns production function--meaning if we double capital (k) and labor (l), the output doubles. It is also known as a CES production function. We will have more to say about CES production functions at a later time.

Given the following production function: decide if it shows:

a: incresaing returns to scale, b: constant returns to scale, and c: decresaing returns to scale.

> Q:=10*k^.30*l^.7;

> eval(Q,{k=7,l=3});

> eval(Q,{k=14,l=6});

The production function displays constant returns to scale because alpha+beta =1.

> Q:=10*k^.40*l^.7;

> eval(Q,{k=7,l=3});

> eval(Q,{k=14,l=6});

> plot3d(Q,k=0..14,l=0..6);

Do Example 12, page. 134. Given the budget constraint of 108, pk=3, and pl=4, and the production function q=K^0.4*l^0.5, find the optimum level of capital and labor that will result in maximum output.

> Q:=(k^0.4)*(l^0.6);

> Const:=3*k+4*l=108;

> L:=Q+lambda*(108-3*k-4*l);

Take first derivative with respect to k, l, and lambda and set equal to zero, and solved for the three unknowns.

> Lk:=diff(L,k);

> Ll:=diff(L,l);

> Llambda:=diff(L,lambda);

> solve({Lk, Ll, Llambda},{k,l,lambda});

> eval(Q,{l = 16.20000000, lambda = .1430969081, k = 14.40000000});

The solutions for the three variables provide the optimum output. No other combinations of capital and labor would yield a higher level of output. Please check the second order condition to make sure that we have a relative maximum.